On the Construction of Cantor like Sets

The International Journal Of Engineering And Science (IJES) || Volume || 4 || Issue || 4 || Pages || PP.49-54|| 2015 || ISSN (e): 2319 – 1813 ISSN (p): 2319 – 1805

On the Construction of Cantor like Sets Dr. S. M. Padhye1 , Satish B. Khobragade2 Associate Professor, HOD, Department Of Mathematics, Shri R.L.T. College Of Science, Akola, Maharashtra, India1 Research Student, Department Of Mathematics, Shri R.L.T. College Of Science, Akola, Maharashtra, India 2

----------------------------------------------------------- ABSTRACT---------------------------------------------------------In this paper we construct a Cantor like set S from any sequence { } with 0< <1 with the help of sequence { } of subsets of [0,1] such that ,m( ) = and S = ∩ with m(S) = .Further ∑ = ∞ if and only if m(S) = 0. Cantor ternary set comes out to be a particular case of construction of Cantor like sets by choosing = for all n. Similarly we can construct Cantor - set by choosing

= for all n. In the construction of Cantor - set the length of remaining closed intervals at each , k = 1,2,3,…………… Also we can construct Cantor - set by choosing

stage are equal to

Here in the construction of Cantor -

= for all n.

set the length of remaining closed intervals at each stage are equal to

, k = 1,2,3,……………

KEY WORDS: - Cantor set, Cantor like sets.

----------------------------------------------------------------------------------------------------------------------------- ---------Date of Submission: 04-April-2015 Date of Accepted: 25-April-2015 ----------------------------------------------------------------------------------------------------------------------------- ---------Lemma 1:- Given any sequence { Proof :- First step: Let

} with 0<

<1,∑

= ∞ if and only if

)= 0

=∞ )= 0 i.e.

Then we have to show that Here we use 1- ≤ 1-

≤

1-

≤

1-

≤

1-

,

)= 0

0≤ <1 i =1,2,3,…………..

≤

Multiplying all these inequalities we get, (1-

) (1-

)…………….(1-

)≤

.

۰۰۰۰۰۰

) ≤

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………………………(1)

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On the Construction of Cantor like Sets )= 0, let > 0 be given. Put M =

To show that Since ∑

=∞ then there is N such that for n ≥ N ═>

.

>M >

> > ………………………(2)

< From equation (1) and (2) we get , ) <

for all n ≥ N.

)= 0

Thus )= 0 < ∞ i.e. ∑

Conversely:-Let

< ∞ is convergent. )≠0.

We show that Let Pn = Since

≥ 0,

) ≠1

we choose N so large that

j and ∑ < ∞ . +

+ ۰۰۰۰۰۰<

--------------------- (3)

Then using induction we prove that for all n ≥ N, (1 - )(1)۰۰۰۰۰(1- ) ≥ [1- ( + ۰۰۰۰۰ + )] For n = N, the inequality is obvious. For n > N If (1 - )(1)۰۰۰۰۰(1- ) ≥ [1- ( + +۰۰۰۰ + )] then (1 - )(1)۰۰۰۰۰(1- ) (1 ) ≥ [1- ( + +۰۰۰۰ + )] (1 ) = [1- ( + + ۰۰۰۰ + )] + ( + + ۰۰۰۰ ) ≥ [1- ( + +۰۰۰۰ + )] Thus (1 - )(1)۰۰۰۰۰(1- ) (1 ) ≥ [1- ( + +۰۰۰۰ + )] By induction the inequality holds for n ≥ N i.e. (1 - )(1)۰۰۰۰۰(1- ) ≥ [1- ( + + ۰۰۰۰۰ + )] for all n ≥ N ………….(4) Now by using equation (3) we get, (1 -

)(1-

)۰۰۰۰۰(1-

) ≥ [1- ] =

(1 -

)(1-

)۰۰۰۰۰(1-

) >

Now for n ≥ N,

,n ≥ N

+

………………………. (5)

=

=

=

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>

,n ≥ N

( From equation (5))

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On the Construction of Cantor like Sets >

,n ≥ N

{

}≥

………………………. (6) ≠0

………………………. (7)

Consider -

=

-

-

=

[ 1- (1-

-

=

≥0 ≥0

{

)]

} is monotonic decreasing and bounded below by {

}=

Pn ≥

.

≥

PN-1

Pn =α PN-1 ,

where α≥ 1/2 ) =α { PN-1} ,

═>

where α≥ 1/2

) is a positive number )≠0 **************************************

Lemma 2 :< 1, n ≥ 1,

If 0 <

=

+

+۰۰۰۰۰۰+

+

+۰۰۰۰۰۰+

and

= (1-

) (1-

)…………….(1-

) then

(1-

)≤

≤ Proof :- Given < 1, n ≥ 1,

0< = (1-

) (1-

≥1–( +

=

)…………….(1+۰۰۰۰۰۰+

)

and

) ( From equation (4) of Lemma 1)

≥11–

≤

…………………………………..(1)

Now, (1-

) (1 +

)=1–

(1-

) (1 +

) <1

(1-

<1

)<

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On the Construction of Cantor like Sets Similarly, (1-

(1-

)<

)<

Multiplying all these equations we get, (1-

) (1-

…………………………(2)

)≤

)…………….(1-

Now ) = 1+( + +۰۰۰+ )+( ) ≥ 1+ ═>

+

+۰۰۰)+(

+۰۰۰۰ ( +۰۰۰۰۰

+ +۰۰۰+

≤

Putting in equation (2) we get, (1-

) (1-

)≤

)…………….(1-

≤ ≤

………………………………(3)

From equation (1) and (3) we get, (1-

)≤

≤

*************************** Corollary 3 :If in addition lim

=

and lim

=

then (1 - ) ≤

≤

.

Proof :-By lemma 2 we get, (1Given lim

=

)≤

≤

and lim

(1 - ) ≤

=

≤

*************************** Preposition 4 :,m(

Given any sequence { } with 0< <1,there is a sequence { )= and S = ∩ is Cantor like set with

m(S) =

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} of subsets of [0,1] such that

.

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On the Construction of Cantor like Sets Proof :Let I = [0,1] First stage : We remove middle open intervals =(

i.e. intervals

,

of length

from [0,1]

). = [0,

The remaining two closed intervals are denoted by with measure ) i.e.m( ) =

] and

=[

,1 ].We get the set

=

)

Second stage : Now we remove two middle open intervals closed intervals

and

=(

,

= [0, =[

,

from the remaining two

) and

).The remaining four closed intervals are denoted by

],

=[

,

, ],

],

=[

Length of removed intervals = m( We get the set

)

of length

=(

i.e. we remove

,1 ]. ) + m(

)=

)

)

<

as union of remaining four closed intervals i.e. i.e.m( ) =

=

with measure

Third stage : Now we remove four middle open intervals of length , , , remaining four closed intervals and i.e. we remove , , ,=

(

,

),

=(

,

),

=(

,

=(

),

,

)

The remaining four closed intervals are denoted by =[

,

=[

=[

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], ,

=[

= [ 0,

=[

],

=[

],

,

],

,

, ,

from the

], ],

],

=[

,1]

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On the Construction of Cantor like Sets Now, Length of removed interval = m(

)+m(

)+m(

) .

<

= We get the set

)+m(

=

with measure i.e. m( ) =

۰۰۰۰۰۰

Continuing in this way we can construct . m(

)=

S4

Then If S =

. , n ≥ 4 of measure

۰۰۰۰۰۰

=

)

.

then m(S) = =

)

************************************* Conclusion :- Using this Cantor like set we can construct different Cantor like sets by varying or fixing

.

REFERENCES:[1] [2] [3] [4]

G.de. Barra, “Measure Theory And Integration”, New Age International (p) Limited, 2003.

Kenneth Falconer, “Fractal Geometry: Mathematical Foundations and Applications”, John Willey and Sons.1990.

Kannan V. “Cantor set: From Classical To Modern”. The Mathematical Student, Vol 63,No.1-4(1994)P.243-257.

E. C. Titchmarsh ,“The Theory Of Functions”, Oxford University Press, Oxford Second Edition -1987.

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